Finite Group Databasebeta

← #28299722

Subgroups — #28299722

Order 13568.

500 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290607C211yesmaximal, characteristic
2#20290570C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5C4×#13133193C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133274—21no—
8C2×#13133274—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133193C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810504—21no—
17#16810494—21no—
18#16810480C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C4×#11719921—21no—
22C2×C4×Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26#13133274—42no—
27#13133274—42no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31#13133193—22no—
32#13133193—22no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43C42×D53—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C4×Q8×C53—42no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53C2×C411yescharacteristic
55C23×D53C22⋊C411yescharacteristic
56C2×Q8×C53—21no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—22no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53SD1612yes—
80C2×C4×D53Q1612yes—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—42no—
89C42×C53—22no—
90C42×C53—42no—
91#11719921—24no—
92#11719921—42no—
93C4⋊Dic53—21no—
94C4⋊Dic53—41no—
95C4⋊Dic53M1612yes—
96#11719919—24no—
97#11719919—42no—
98C4×Dic53—22no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C4×Dic53—42no—
102C23×C53C2×C22⋊C411yescharacteristic
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53#10312yes—
110Q8×C53—42no—
111Q8×C53—42no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#9912yes—
125C2×C4×C53#10312yes—
126C2×C4×C53C2×M1611yescharacteristic
127C2×C4×C53C2×M1611yescharacteristic
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—22no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53C2×SD1611yescharacteristic
142C2×Dic53C2×Q1611yescharacteristic
143C2×Dic53#13512yes—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—44no—
147C4×D53—42no—
148C4×D53—42no—
149#88324—82no—
150#88324—82no—
151#7648—531nomaximal, Sylow
152C22×C53#40711yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53#41311yescharacteristic
155C22×C53C2×#10311yescharacteristic
156C22×C53#41611yescharacteristic
157C22×C53#42111yescharacteristic
158C22×C53#42111yescharacteristic
159C2×D53—22no—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—22no—
170C4×C53—42no—
171C4×C53—81no—
172Dic53—22no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—42no—
177Dic53—81no—
178C4×C22⋊Q8—531no—
179#1817—531no—
180#1797—531no—
181#1272—532no—
182#1271—532no—
183C2×C53#146511yescharacteristic
184C2×C53#148711yescharacteristic
185C2×C53#148811yescharacteristic
186C2×C53#151611yescharacteristic
187C2×C53#152011yescharacteristic
188C2×C53#153511yescharacteristic
189C2×C53#154211yescharacteristic
190D53—42no—
191C2×C22⋊Q8—531no—
192C2×C4×Q8#615711yescharacteristic
193C22×C42—531no—
194C2×C4⋊C8—1061no—
195C2×C22⋊C8—1061no—
196C2×C22⋊C8—1061no—
197#391—531no—
198#389—531no—
199#387—1061no—
200#385—1061no—
201#380—531no—
202C4×C4⋊C4—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—1061no—
205#336—532no—
206#336—532no—
207#336—1061no—
208C53#764811yescharacteristic, complemented, Sylow
209C22×Q8C2×#615711yescharacteristic
210C23×C4—531no—
211C23×C4—531no—
212C23×C4—531no—
213C22×C8—1062no—
214C22×C8—1062no—
215C22⋊Q8—1062no—
216C22⋊Q8—1062no—
217C4×Q8—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4C2×#615711yescharacteristic
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C42—21no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—1061no—
231C2×C42—2122no—
232C2×C42C2×#615711yescharacteristic
233C4⋊C8—2122no—
234C22⋊C8—2122no—
235C22⋊C8—2122no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24—531no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#1171993612yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—532no—
251C22×C4—532no—
252C22×C4—531no—
253C22×C4—1062no—
254C22×C4—1061no—
255C22×C4—1062no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4C22⋊#615711yescharacteristic
265C22×C4C22⋊#615711yescharacteristic
266C22×C4C22×#615711yescharacteristic
267C2×C8—2122no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—532no—
276C4⋊C4—1064no—
277C4⋊C4—1061no—
278C4⋊C4—2121no—
279C4⋊C4—2122no—
280C22⋊C4—1064no—
281C22⋊C4—2122no—
282C42—22no—
283C42—42no—
284C42—1062no—
285C42—2121no—
286C42—2121no—
287C42—2122no—
288C23—532no—
289C23—1061no—
290C23—1061no—
291C23—1061no—
292C23—1061no—
293C23—1061no—
294C23—1061no—
295C23C2×C22⋊#615711yescharacteristic
296Q8—42no—
297Q8—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—532no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1062no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2124no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4#1313327512yes—
332C2×C4#1313328012yes—
333C2×C4D53⋊M1611yescharacteristic
334C2×C4C2×#1171993611yescharacteristic
335C8—4242no—
336C8—4242no—
337C22—1062no—
338C22—1062no—
339C22—1062no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22#1681159411yescharacteristic
345C22C2×#1313327511yescharacteristic
346C22#1681160911yescharacteristic
347C22C2×#1313328011yescharacteristic
348C22#1681161311yescharacteristic
349C22#1681162011yescharacteristic
350C22#1681162211yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—81no—
357C4—1062no—
358C4—1061no—
359C4—1061no—
360C4—4241no—
361C4—2122no—
362C4—2121no—
363C2—2122no—
364C2#2028997011yescharacteristic
365C2#2029004411yescharacteristic
366C2#2029004911yescharacteristic
367C2#2029014111yescharacteristic
368C2#2029015511yescharacteristic
369C2#2029020111yescharacteristic
370C2#2029021611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7648
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289534C2semidirect
208. C53#7648semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202906072
#202905702
#202895352
#202895342
C4×#131331932
#764853